### Subspaces and Cosets of Subspaces in Real Linear Space

by
Wojciech A. Trybulec

Copyright (c) 1989 Association of Mizar Users

MML identifier: RLSUB_1
[ MML identifier index ]

```environ

vocabulary RLVECT_1, BOOLE, ARYTM_1, RELAT_1, FUNCT_1, BINOP_1, RLSUB_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, REAL_1, MCART_1,
FUNCT_1, RELSET_1, FUNCT_2, DOMAIN_1, BINOP_1, STRUCT_0, RLVECT_1;
constructors REAL_1, DOMAIN_1, RLVECT_1, PARTFUN1, MEMBERED, XBOOLE_0;
clusters FUNCT_1, RLVECT_1, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1,
XBOOLE_0;
requirements NUMERALS, BOOLE, SUBSET, ARITHM;
definitions RLVECT_1, TARSKI, XBOOLE_0;
theorems FUNCT_1, FUNCT_2, RLVECT_1, TARSKI, ZFMISC_1, RELAT_1, RELSET_1,
XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1;
schemes XBOOLE_0;

begin

reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,b for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for set;

::
::  Introduction of predicate lineary closed subsets of the carrier.
::

definition let V; let V1;
attr V1 is lineary-closed means
:Def1:
(for v,u st v in V1 & u in V1 holds v + u in V1) &
(for a,v st v in V1 holds a * v in V1);
end;

canceled 3;

theorem Th4:
V1 <> {} & V1 is lineary-closed implies 0.V in V1
proof assume that A1: V1 <> {} and A2: V1 is lineary-closed;
consider x being Element of V1;
reconsider x as Element of V by A1,TARSKI:def 3;
0 * x in V1 by A1,A2,Def1;
hence thesis by RLVECT_1:23;
end;

theorem Th5:
V1 is lineary-closed implies (for v st v in V1 holds - v in V1)
proof assume A1: V1 is lineary-closed;
let v; assume v in V1;
then (- 1) * v in V1 by A1,Def1;
hence thesis by RLVECT_1:29;
end;

theorem
V1 is lineary-closed implies
(for v,u st v in V1 & u in V1 holds v - u in V1)
proof assume A1: V1 is lineary-closed;
let v,u; assume that A2: v in V1 and A3: u in V1;
v - u = v + (- u) & - u in V1 by A1,A3,Th5,RLVECT_1:def 11;
hence thesis by A1,A2,Def1;
end;

theorem Th7:
{0.V} is lineary-closed
proof
thus for v,u st v in {0.V} & u in {0.V} holds v + u in {0.V}
proof let v,u;
assume v in {0.V} & u in {0.V};
then v = 0.V & u = 0.V by TARSKI:def 1;
then v + u = 0.V & 0.V in {0.V} by RLVECT_1:10,TARSKI:def 1;
hence thesis;
end;
let a,v;
assume A1: v in {0.V};
then v = 0.V by TARSKI:def 1;
hence thesis by A1,RLVECT_1:23;
end;

theorem
the carrier of V = V1 implies V1 is lineary-closed
proof assume A1: the carrier of V = V1;
hence for v,u st v in V1 & u in V1 holds v + u in V1;
let a,v;
assume v in V1;
thus a * v in V1 by A1;
end;

theorem
V1 is lineary-closed & V2 is lineary-closed &
V3 = {v + u : v in V1 & u in V2} implies V3 is lineary-closed
proof assume that A1: V1 is lineary-closed & V2 is lineary-closed and
A2: V3 = {v + u : v in V1 & u in V2};
thus for v,u st v in V3 & u in V3 holds v + u in V3
proof let v,u;
assume that A3: v in V3 and A4: u in V3;
consider v1,v2 such that A5: v = v1 + v2 and
A6: v1 in V1 & v2 in V2 by A2,A3;
consider u1,u2 such that A7: u = u1 + u2 and
A8: u1 in V1 & u2 in V2 by A2,A4;
A9: v1 + u1 in V1 & v2 + u2 in V2 by A1,A6,A8,Def1;
v + u = ((v1 + v2) + u1) + u2 by A5,A7,RLVECT_1:def 6
.= ((v1 + u1) + v2) + u2 by RLVECT_1:def 6
.= (v1 + u1) + (v2 + u2) by RLVECT_1:def 6;
hence thesis by A2,A9;
end;
let a,v;
assume v in V3;
then consider v1,v2 such that A10: v = v1 + v2 and
A11: v1 in V1 & v2 in V2 by A2;
A12: a * v1 in V1 & a * v2 in V2 by A1,A11,Def1;
a * v = a * v1 + a * v2 by A10,RLVECT_1:def 9;
hence a * v in V3 by A2,A12;
end;

theorem
V1 is lineary-closed & V2 is lineary-closed implies
V1 /\ V2 is lineary-closed
proof assume A1: V1 is lineary-closed & V2 is lineary-closed;
thus for v,u st v in V1 /\ V2 & u in V1 /\ V2 holds v + u in V1 /\ V2
proof let v,u;
assume v in V1 /\ V2 & u in V1 /\ V2;
then v in V1 & v in V2 & u in V1 & u in V2 by XBOOLE_0:def 3;
then v + u in V1 & v + u in V2 by A1,Def1;
hence thesis by XBOOLE_0:def 3;
end;
let a,v;
assume v in V1 /\ V2;
then v in V1 & v in V2 by XBOOLE_0:def 3;
then a * v in V1 & a * v in V2 by A1,Def1;
hence thesis by XBOOLE_0:def 3;
end;

definition let V;
mode Subspace of V -> RealLinearSpace means
:Def2:
the carrier of it c= the carrier of V &
the Zero of it = the Zero of V &
the add of it = (the add of V) | [:the carrier of it,the carrier of it:] &
the Mult of it = (the Mult of V) | [:REAL, the carrier of it:];
existence
proof
the carrier of V c= the carrier of V &
the Zero of V = the Zero of V &
the add of V = (the add of V) | [:the carrier of V,the carrier of V:] &
the Mult of V = (the Mult of V) | [:REAL, the carrier of V:] by FUNCT_2:40
;
hence thesis;
end;
end;

reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;

::
::  Axioms of the subspaces of real linear spaces.
::

canceled 5;

theorem
x in W1 & W1 is Subspace of W2 implies x in W2
proof assume x in W1 & W1 is Subspace of W2;
then x in the carrier of W1 & the carrier of W1 c= the carrier of W2
by Def2,RLVECT_1:def 1
;
hence thesis by RLVECT_1:def 1;
end;

theorem Th17:
x in W implies x in V
proof assume x in W;
then x in the carrier of W & the carrier of W c= the carrier of V
by Def2,RLVECT_1:def 1
;
hence thesis by RLVECT_1:def 1;
end;

theorem Th18:
w is VECTOR of V
proof w in W by RLVECT_1:3;
then w in V by Th17;
hence thesis by RLVECT_1:def 1;
end;

theorem Th19:
0.W = 0.V
proof
thus 0.W = the Zero of W by RLVECT_1:def 2
.= the Zero of V by Def2
.= 0.V by RLVECT_1:def 2;
end;

theorem
0.W1 = 0.W2
proof
thus 0.W1 = 0.V by Th19
.= 0.W2 by Th19;
end;

theorem Th21:
w1 = v & w2 = u implies w1 + w2 = v + u
proof assume A1: v = w1 & u = w2;
reconsider ww1 = w1, ww2 = w2 as VECTOR of V by Th18;
A2: v + u = (the add of V).[ww1,ww2] by A1,RLVECT_1:def 3;
w1 + w2 = (the add of W).[w1,w2] by RLVECT_1:def 3
.= ((the add of V) | [:the carrier of W, the carrier of W:]
).[w1,w2] by Def2;
hence thesis by A2,FUNCT_1:72;
end;

theorem Th22:
w = v implies a * w = a * v
proof assume A1: w = v;
reconsider ww1 = w as VECTOR of V by Th18;
A2: a * v = (the Mult of V).[a,ww1] by A1,RLVECT_1:def 4;
a * w = (the Mult of W).[a,w] by RLVECT_1:def 4
.= ((the Mult of V) | [:REAL, the carrier of W:]).[a,w] by Def2;
hence thesis by A2,FUNCT_1:72;
end;

theorem Th23:
w = v implies - v = - w
proof assume A1: w = v;
- v = (- 1) * v & - w = (- 1) * w by RLVECT_1:29;
hence thesis by A1,Th22;
end;

theorem Th24:
w1 = v & w2 = u implies w1 - w2 = v - u
proof assume that A1: w1 = v and A2: w2 = u;
A3: - w2 = - u by A2,Th23;
w1 - w2 = w1 + (- w2) & v - u = v + (- u) by RLVECT_1:def 11;
hence thesis by A1,A3,Th21;
end;

Lm1: the carrier of W = V1 implies V1 is lineary-closed
proof assume A1: the carrier of W = V1;
set VW = the carrier of W;
reconsider WW = W as RealLinearSpace;
thus for v,u st v in V1 & u in V1 holds v + u in V1
proof let v,u;
assume v in V1 & u in V1;
then reconsider vv = v, uu = u as VECTOR of WW by A1;
reconsider vw = vv + uu as Element of VW;
vw in V1 by A1;
hence v + u in V1 by Th21;
end;
let a,v;
assume v in V1;
then reconsider vv = v as VECTOR of WW by A1;
reconsider vw = a * vv as Element of VW;
vw in V1 by A1;
hence a * v in V1 by Th22;
end;

theorem Th25:
0.V in W
proof 0.W in W & 0.V = 0.W by Th19,RLVECT_1:3;
hence thesis;
end;

theorem
0.W1 in W2
proof 0.W1 = 0.V by Th19;
hence thesis by Th25;
end;

theorem
0.W in V
proof 0.W in W by RLVECT_1:3;
hence thesis by Th17;
end;

theorem Th28:
u in W & v in W implies u + v in W
proof assume u in W & v in W;
then A1: u in the carrier of W & v in the carrier of W by RLVECT_1:def 1;
reconsider VW = the carrier of W as Subset of V by Def2;
VW is lineary-closed by Lm1;
then u + v in the carrier of W by A1,Def1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th29:
v in W implies a * v in W
proof assume v in W;
then A1: v in the carrier of W by RLVECT_1:def 1;
reconsider VW = the carrier of W as Subset of V by Def2;
VW is lineary-closed by Lm1;
then a * v in the carrier of W by A1,Def1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th30:
v in W implies - v in W
proof assume v in W;
then (- 1) * v in W by Th29;
hence thesis by RLVECT_1:29;
end;

theorem Th31:
u in W & v in W implies u - v in W
proof assume that A1: u in W and A2: v in W;
- v in W by A2,Th30;
then u + (- v) in W by A1,Th28;
hence thesis by RLVECT_1:def 11;
end;

reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;

theorem Th32:
V1 = D &
d1 = 0.V &
A = (the add of V) | [:V1,V1:] &
M = (the Mult of V) | [:REAL,V1:] implies
RLSStruct (# D,d1,A,M #) is Subspace of V
proof assume that A1: V1 = D and A2: d1 = 0.V and
A3: A = (the add of V) | [:V1,V1:] and
A4: M = (the Mult of V) | [:REAL,V1:];
set W = RLSStruct (# D,d1,A,M #);
A5: the Zero of W = the Zero of V by A2,RLVECT_1:def 2;
A6: for x,y being VECTOR of W holds x + y = (the add of V).[x,y]
proof let x,y be VECTOR of W;
x + y = ((the add of V) | [:the carrier of W, the carrier of W:]
).[x,y] by A1,A3,RLVECT_1:def 3;
hence thesis by FUNCT_1:72;
end;
A7: for a for x being VECTOR of W holds a * x = (the Mult of V).[a,x]
proof let a; let x be VECTOR of W;
a * x = ((the Mult of V) | [:REAL, the carrier of W:]
).[a,x] by A1,A4,RLVECT_1:def 4;
hence thesis by FUNCT_1:72;
end;
A8: d1 = 0.W by RLVECT_1:def 2;
W is Abelian add-associative right_zeroed right_complementable
RealLinearSpace-like
proof
set AV = the add of V; set MV = the Mult of V;
thus for x,y being VECTOR of W holds x + y = y + x
proof let x,y be VECTOR of W;
reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
thus x + y = AV.[x1,y1] by A6
.= y1 + x1 by RLVECT_1:def 3
.= AV.[y1,x1] by RLVECT_1:def 3
.= y + x by A6;
end;
thus for x,y,z being VECTOR of W holds (x + y) + z = x + (y + z)
proof let x,y,z be VECTOR of W;
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1,TARSKI:def 3;
thus (x + y) + z = AV.[x + y,z1] by A6
.= AV.[AV.[x1,y1],z1] by A6
.= AV.[x1 + y1,z1] by RLVECT_1:def 3
.= (x1 + y1) + z1 by RLVECT_1:def 3
.= x1 + (y1 + z1) by RLVECT_1:def 6
.= AV.[x1,y1 + z1] by RLVECT_1:def 3
.= AV.[x1,AV.[y1,z1]] by RLVECT_1:def 3
.= AV.[x1,y + z] by A6
.= x + (y + z) by A6;
end;
thus for x being VECTOR of W holds x + 0.W = x
proof let x be VECTOR of W;
reconsider y = x, z = 0.W as VECTOR of V by A1,TARSKI:def 3;
thus x + 0.W = AV.[y,z] by A6
.= y + 0.V by A2,A8,RLVECT_1:def 3
.= x by RLVECT_1:10;
end;
thus for x being VECTOR of W
ex y being VECTOR of W st x + y = 0.W
proof let x be VECTOR of W;
reconsider x1 = x as VECTOR of V by A1,TARSKI:def 3;
consider v such that A9: x1 + v = 0.V by RLVECT_1:def 8;
v = - x1 by A9,RLVECT_1:def 10
.= (- 1) * x1 by RLVECT_1:29
.= MV.[- 1,x1] by RLVECT_1:def 4
.= (- 1) * x by A7;
then reconsider y = v as VECTOR of W;
take y;
thus x + y = AV.[x1,v] by A6
.= 0.W by A2,A8,A9,RLVECT_1:def 3;
end;
thus for a
for x,y being VECTOR of W holds a * (x + y) = a * x + a * y
proof let a; let x,y be VECTOR of W;
reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
thus a * (x + y) = MV.[a,x + y] by A7
.= MV.[a,AV.[x1,y1]] by A6
.= MV.[a,x1 + y1] by RLVECT_1:def 3
.= a * (x1 + y1) by RLVECT_1:def 4
.= a * x1 + a * y1 by RLVECT_1:def 9
.= AV.[a * x1,a * y1] by RLVECT_1:def 3
.= AV.[MV.[a,x1],a * y1] by RLVECT_1:def 4
.= AV.[MV.[a,x1],MV.[a,y1]] by RLVECT_1:def 4
.= AV.[MV.[a,x1],a * y] by A7
.= AV.[a * x, a * y] by A7
.= a * x + a * y by A6;
end;
thus for a,b
for x being VECTOR of W holds (a + b) * x = a * x + b * x
proof let a,b; let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
thus (a + b) * x = MV.[a + b,y] by A7
.= (a + b) * y by RLVECT_1:def 4
.= a * y + b * y by RLVECT_1:def 9
.= AV.[a * y,b * y] by RLVECT_1:def 3
.= AV.[MV.[a,y],b * y] by RLVECT_1:def 4
.= AV.[MV.[a,y],MV.[b,y]] by RLVECT_1:def 4
.= AV.[MV.[a,y],b * x] by A7
.= AV.[a * x,b * x] by A7
.= a * x + b * x by A6;
end;
thus for a,b
for x being VECTOR of W holds (a * b) * x = a * (b * x)
proof let a,b; let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
thus (a * b) * x = MV.[(a * b),y] by A7
.= (a * b) * y by RLVECT_1:def 4
.= a * (b * y) by RLVECT_1:def 9
.= MV.[a,b * y] by RLVECT_1:def 4
.= MV.[a,MV.[b,y]] by RLVECT_1:def 4
.= MV.[a,b * x] by A7
.= a * (b * x) by A7;
end;
let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
thus 1 * x = MV.[1,y] by A7
.= 1 * y by RLVECT_1:def 4
.= x by RLVECT_1:def 9;
end;
hence thesis by A1,A3,A4,A5,Def2;
end;

theorem Th33:
V is Subspace of V
proof
thus the carrier of V c= the carrier of V &
the Zero of V = the Zero of V;
thus thesis by FUNCT_2:40;
end;

theorem Th34:
for V,X being strict RealLinearSpace holds
V is Subspace of X & X is Subspace of V implies V = X
proof let V,X be strict RealLinearSpace;
assume A1: V is Subspace of X & X is Subspace of V;
set VV = the carrier of V; set VX = the carrier of X;
set AV = the add of V; set AX = the add of X;
set MV = the Mult of V; set MX = the Mult of X;
VV c= VX & VX c= VV by A1,Def2;
then A2: VV = VX by XBOOLE_0:def 10;
A3: the Zero of V = the Zero of X by A1,Def2;
AV = AX | [:VV,VV:] & AX = AV | [:VX,VX:] by A1,Def2;
then A4: AV = AX by A2,RELAT_1:101;
MV = MX | [:REAL,VV:] & MX = MV | [:REAL,VX:] by A1,Def2;
hence thesis by A2,A3,A4,RELAT_1:101;
end;

theorem Th35:
V is Subspace of X & X is Subspace of Y implies V is Subspace of Y
proof assume A1: V is Subspace of X & X is Subspace of Y;
thus the carrier of V c= the carrier of Y
proof
the carrier of V c= the carrier of X &
the carrier of X c= the carrier of Y by A1,Def2;
hence thesis by XBOOLE_1:1;
end;
thus the Zero of V = the Zero of Y
proof the Zero of V = the Zero of X & the Zero of X = the Zero of Y
by A1,Def2;
hence thesis;
end;
thus the add of V = (the add of Y) | [:the carrier of V, the carrier of V:]
proof set AV = the add of V; set VV = the carrier of V;
set AX = the add of X; set VX = the carrier of X;
set AY = the add of Y;
AV = AX | [:VV,VV:] & AX = AY | [:VX,VX:] & VV c= VX by A1,Def2;
then AV = (AY | [:VX,VX:]) | [:VV,VV:] & [:VV,VV:] c= [:VX,VX:]
by ZFMISC_1:119;
hence thesis by FUNCT_1:82;
end;
set MV = the Mult of V; set VV = the carrier of V;
set MX = the Mult of X; set VX = the carrier of X;
set MY = the Mult of Y;
MV = MX | [:REAL,VV:] & MX = MY | [:REAL,VX:] & VV c= VX by A1,Def2;
then MV = (MY | [:REAL,VX:]) | [:REAL,VV:] & [:REAL,VV:] c= [:REAL,VX:]
by ZFMISC_1:118;
hence thesis by FUNCT_1:82;
end;

theorem Th36:
the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2
proof assume A1: the carrier of W1 c= the carrier of W2;
set VW1 = the carrier of W1; set VW2 = the carrier of W2;
set AV = the add of V; set MV = the Mult of V;
the Zero of W1 = the Zero of V & the Zero of W2 = the Zero of V by Def2;
hence the carrier of W1 c= the carrier of W2 &
the Zero of W1 = the Zero of W2 by A1;
thus the add of W1 =
(the add of W2) | [:the carrier of W1,the carrier of W1:]
proof
the add of W1 = AV | [:VW1,VW1:] & the add of W2 = AV | [:VW2,VW2:] &
[:VW1,VW1:] c= [:VW2,VW2:] by A1,Def2,ZFMISC_1:119;
hence thesis by FUNCT_1:82;
end;
the Mult of W1 = MV | [:REAL,VW1:] & the Mult of W2 = MV | [:REAL,VW2:]
&
[:REAL,VW1:] c= [:REAL,VW2:] by A1,Def2,ZFMISC_1:118;
hence thesis by FUNCT_1:82;
end;

theorem
(for v st v in W1 holds v in W2) implies W1 is Subspace of W2
proof assume A1: for v st v in W1 holds v in W2;
the carrier of W1 c= the carrier of W2
proof let x be set;
assume A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2,RLVECT_1:def 1;
then v in W2 by A1;
hence thesis by RLVECT_1:def 1;
end;
hence thesis by Th36;
end;

definition let V;
cluster strict Subspace of V;
existence
proof
the carrier of V is Subset of V iff
the carrier of V c= the carrier of V;
then reconsider V1 = the carrier of V as Subset of V;
the Zero of V = 0.V &
the add of V = (the add of V) | [:V1,V1:] &
the Mult of V = (the Mult of V) | [:REAL,V1:] by FUNCT_2:40,RLVECT_1:def 2
;
then RLSStruct(#the carrier of V,the Zero of V,the add of V,the Mult of V
#)
is Subspace of V by Th32;
hence thesis;
end;
end;

theorem Th38:
for W1,W2 being strict Subspace of V holds
the carrier of W1 = the carrier of W2 implies W1 = W2
proof let W1,W2 be strict Subspace of V;
assume the carrier of W1 = the carrier of W2;
then W1 is Subspace of W2 & W2 is Subspace of W1 by Th36;
hence thesis by Th34;
end;

theorem Th39:
for W1,W2 being strict Subspace of V holds
(for v holds v in W1 iff v in W2) implies W1 = W2
proof let W1,W2 be strict Subspace of V;
assume A1: for v holds v in W1 iff v in W2;
x in the carrier of W1 iff x in the carrier of W2
proof
thus x in the carrier of W1 implies x in the carrier of W2
proof assume A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2,RLVECT_1:def 1;
then v in W2 by A1;
hence thesis by RLVECT_1:def 1;
end;
assume A3: x in the carrier of W2;
the carrier of W2 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A3;
v in W2 by A3,RLVECT_1:def 1;
then v in W1 by A1;
hence thesis by RLVECT_1:def 1;
end;
then the carrier of W1 = the carrier of W2 by TARSKI:2;
hence thesis by Th38;
end;

theorem
for V being strict RealLinearSpace, W being strict Subspace of V holds
the carrier of W = the carrier of V implies W = V
proof let V be strict RealLinearSpace, W be strict Subspace of V;
assume A1: the carrier of W = the carrier of V;
V is Subspace of V by Th33;
hence thesis by A1,Th38;
end;

theorem
for V being strict RealLinearSpace, W being strict Subspace of V holds
(for v being VECTOR of V holds v in W iff v in V) implies W = V
proof let V be strict RealLinearSpace, W be strict Subspace of V;
assume A1: for v being VECTOR of V holds v in W iff v in V;
V is Subspace of V by Th33;
hence thesis by A1,Th39;
end;

theorem
the carrier of W = V1 implies V1 is lineary-closed by Lm1;

theorem Th43:
V1 <> {} & V1 is lineary-closed implies
(ex W being strict Subspace of V st V1 = the carrier of W)
proof assume that A1: V1 <> {} and A2: V1 is lineary-closed;
reconsider D = V1 as non empty set by A1;
reconsider d1 = 0.V as Element of D by A2,Th4;
set A = (the add of V) | [:V1,V1:];
set M = (the Mult of V) | [:REAL,V1:];
set VV = the carrier of V;
dom(the add of V) = [:VV,VV:] by FUNCT_2:def 1;
then dom A = [:VV,VV:] /\ [:V1,V1:] & [:V1,V1:] c= [:VV,VV:]
by RELAT_1:90;
then A3: dom A = [:D,D:] by XBOOLE_1:28;
dom(the Mult of V) = [:REAL,VV:] by FUNCT_2:def 1;
then dom M = [:REAL,VV:] /\ [:REAL,V1:] & [:REAL,V1:] c= [:REAL,VV:]
by RELAT_1:90,ZFMISC_1:118;
then A4: dom M = [:REAL,D:] by XBOOLE_1:28;
A5:        D = rng A
proof
now let y be set;
thus y in D implies ex x being set st x in dom A & y = A.x
proof assume A6: y in D;
then reconsider v1 = y, v0 = d1 as Element of VV;

A7: [d1,y] in [:D,D:] & [d1,y] in [:VV,VV:]
by A6,ZFMISC_1:106;
then A.[d1,y] = (the add of V).[d1,y] by FUNCT_1:72
.= v0 + v1 by RLVECT_1:def 3
.= y by RLVECT_1:10;
hence thesis by A3,A7;
end;
given x being set such that A8: x in dom A and A9: y = A.x;
consider x1,x2 being set such that A10: x1 in D & x2 in D
and A11: x = [x1,x2]
by A3,A8,ZFMISC_1:def 2;
A12: [x1,x2] in [:VV,VV:] & [x1,x2] in [:V1,V1:]
by A10,ZFMISC_1:106;
reconsider v1 = x1, v2 = x2 as Element of VV by A10;

y = (the add of V).[x1,x2] by A9,A11,A12,FUNCT_1:72
.= v1 + v2 by RLVECT_1:def 3;
hence y in D by A2,A10,Def1;
end;
hence thesis by FUNCT_1:def 5;
end;

A13:        D = rng M
proof
now let y be set;
thus y in D implies ex x being set st x in dom M & y = M.x
proof assume A14: y in D;
then reconsider v1 = y as Element of VV;

A15: [1,y] in [:REAL,D:] & [1,y] in [:REAL,VV:]
by A14,ZFMISC_1:106;
then M.[1,y] = (the Mult of V).[1,y] by FUNCT_1:72
.= 1 * v1 by RLVECT_1:def 4
.= y by RLVECT_1:def 9;
hence thesis by A4,A15;
end;
given x being set such that A16: x in dom M and A17: y = M.x;
consider x1,x2 being set such that A18: x1 in REAL and A19: x2 in
D
and A20: x = [x1,x2]
by A4,A16,ZFMISC_1:def 2;
A21: [x1,x2] in [:REAL,VV:] & [x1,x2] in [:REAL,V1:]
by A18,A19,ZFMISC_1:106;
reconsider v2 = x2 as Element of VV by A19;

reconsider xx1 = x1 as Real by A18;
y = (the Mult of V).[x1,x2] by A17,A20,A21,FUNCT_1:72
.= xx1 * v2 by RLVECT_1:def 4;
hence y in D by A2,A19,Def1;
end;
hence thesis by FUNCT_1:def 5;
end;
reconsider A as Function of [:D,D:],D by A3,A5,FUNCT_2:def 1,RELSET_1:11;
reconsider M as Function of [:REAL,D:],D
by A4,A13,FUNCT_2:def 1,RELSET_1:11;
set W = RLSStruct (# D,d1,A,M #);
W is Subspace of V & the carrier of W = D by Th32;
hence thesis;
end;

::
::  Definition of zero subspace and improper subspace of real linear space.
::

definition let V;
func (0).V -> strict Subspace of V means
:Def3: the carrier of it = {0.V};
correctness
proof {0.V} is lineary-closed & {0.V} <> {} by Th7;
hence thesis by Th38,Th43;
end;
end;

definition let V;
func (Omega).V -> strict Subspace of V equals
:Def4: the RLSStruct of V;
coherence
proof set W = the RLSStruct of V;
W is Abelian add-associative right_zeroed right_complementable
RealLinearSpace-like
proof
A1:     0.W = the Zero of W by RLVECT_1:def 2 .= 0.V by RLVECT_1:def 2;
A2:     now let a; let v,w be VECTOR of W, v',w' be VECTOR of V such that
A3:       v=v' & w=w';
thus v+w = (the add of W).[v,w] by RLVECT_1:def 3
.= v'+w' by A3,RLVECT_1:def 3;
thus a*v = (the Mult of W).[a,v] by RLVECT_1:def 4
.= a*v' by A3,RLVECT_1:def 4;
end;
thus for v,w being VECTOR of W holds v + w = w + v
proof let v,w be VECTOR of W;
reconsider v'=v,w'=w as VECTOR of V;
thus v + w = w' + v' by A2 .= w + v by A2;
end;
thus for u,v,w being VECTOR of W holds (u + v) + w = u + (v + w)
proof let u,v,w be VECTOR of W;
reconsider u'=u,v'=v,w'=w as VECTOR of V;
A4:        v + w = v' + w' & u + v = u' + v' by A2;
hence (u + v) + w = (u' + v') + w' by A2
.= u' + (v' + w') by RLVECT_1:def 6
.= u + (v + w) by A2,A4;
end;
thus for v being VECTOR of W holds v + 0.W = v
proof let v be VECTOR of W;
reconsider v'=v as VECTOR of V;
thus v + 0.W = v' + 0.V by A1,A2 .= v by RLVECT_1:10;
end;
thus for v being VECTOR of W ex w being VECTOR of W st v + w = 0.W
proof let v be VECTOR of W;
reconsider v'=v as VECTOR of V;
consider w' being VECTOR of V such that
A5:         v' + w' = 0.V by RLVECT_1:def 8;
reconsider w=w' as VECTOR of W;
take w;
thus v + w = 0.W by A1,A2,A5;
end;
thus for a for v,w being VECTOR of W holds a * (v + w) = a * v + a * w
proof let a; let v,w be VECTOR of W;
reconsider v'=v,w'=w as VECTOR of V;
A6:        v + w = v' + w' & a * v = a * v' & a * w = a * w' by A2;
hence a * (v + w) = a * (v' + w') by A2
.= a * v' + a * w' by RLVECT_1:def 9
.= a * v + a * w by A2,A6;
end;
thus for a,b for v being VECTOR of W holds (a + b) * v = a * v + b * v
proof let a,b; let v be VECTOR of W;
reconsider v'=v as VECTOR of V;
A7:        a * v = a * v' & b * v = b * v' by A2;
thus (a + b) * v = (a + b) * v' by A2
.= a * v' + b * v' by RLVECT_1:def 9
.= a * v + b * v by A2,A7;
end;
thus for a,b for v being VECTOR of W holds (a * b) * v = a * (b * v)
proof let a,b; let v be VECTOR of W;
reconsider v'=v as VECTOR of V;
A8:        b * v = b * v' by A2;
thus (a * b) * v = (a * b) * v' by A2
.= a * (b * v') by RLVECT_1:def 9
.= a * (b * v) by A2,A8;
end;
thus for v being VECTOR of W holds 1 * v = v
proof let v be VECTOR of W;
reconsider v'=v as VECTOR of V;
thus 1 * v = 1 * v' by A2 .= v by RLVECT_1:def 9;
end;
end;
then reconsider W as RealLinearSpace;
W is Subspace of V
proof
thus the carrier of W c= the carrier of V &
the Zero of W = the Zero of V;
thus thesis by FUNCT_2:40;
end;
hence thesis;
end;
end;

::
::  Definitional theorems of zero subspace and improper subspace.
::

canceled 4;

theorem Th48:
(0).W = (0).V
proof the carrier of (0).W = {0.W} & the carrier of (0).V = {0.V} by Def3;
then the carrier of (0).W = the carrier of (0).V & (0).W is Subspace of V
by Th19,
Th35;
hence thesis by Th38;
end;

theorem Th49:
(0).W1 = (0).W2
proof (0).W1 = (0).V & (0).W2 = (0).V by Th48;
hence thesis;
end;

theorem
(0).W is Subspace of V by Th35;

theorem
(0).V is Subspace of W
proof the carrier of (0).V = {0.V} by Def3
.= {0.W} by Th19
.= {the Zero of W} by RLVECT_1:def 2;
hence thesis by Th36;
end;

theorem
(0).W1 is Subspace of W2
proof (0).W1 = (0).W2 & (0).W2 is Subspace of W2 by Th49;
hence thesis;
end;

canceled;

theorem
for V being strict RealLinearSpace holds V is Subspace of (Omega).V
proof let V be strict RealLinearSpace;
V is Subspace of V by Th33;
hence thesis by Def4;
end;

::
::  Introduction of the cosets of subspace.
::

definition let V; let v,W;
func v + W -> Subset of V equals
:Def5: {v + u : u in W};
coherence
proof
defpred P[set] means
ex u st \$1 = v + u & u in W;
consider X being set such that
A1: for x being set holds x in X iff x in the carrier of V &
P[x] from Separation;
X c= the carrier of V
proof let x be set;
assume x in X;
hence x in the carrier of V by A1;
end;
then reconsider X as Subset of V;
set Y = {v + u : u in W};
X = Y
proof
thus X c= Y
proof let x be set;
assume x in X;
then ex u st x = v + u & u in W by A1;
hence thesis;
end;
thus Y c= X
proof let x be set;
assume x in Y;
then ex u st x = v + u & u in W;
hence thesis by A1;
end;
end;
hence thesis;
end;
end;

Lm2: 0.V + W = the carrier of W
proof set A = {0.V + u : u in W};
A1: 0.V + W = A by Def5;
A2: A c= the carrier of W
proof let x be set;
assume x in A;
then consider u such that A3: x = 0.V + u and A4: u in W;
x = u by A3,RLVECT_1:10;
hence thesis by A4,RLVECT_1:def 1;
end;
the carrier of W c= A
proof let x be set;
assume x in the carrier of W;
then A5: x in W by RLVECT_1:def 1;
then x in V by Th17;
then reconsider y = x as Element of V by RLVECT_1:def 1;

0.V + y = x by RLVECT_1:10;
hence thesis by A5;
end;
hence thesis by A1,A2,XBOOLE_0:def 10;
end;

definition let V; let W;
mode Coset of W -> Subset of V means
:Def6: ex v st it = v + W;
existence
proof
reconsider VW = the carrier of W as Subset of V by Def2;
take VW; take 0.V;
thus thesis by Lm2;
end;
end;

reserve B,C for Coset of W;

::
::  Definitional theorems of the cosets.
::

canceled 3;

theorem Th58:
0.V in v + W iff v in W
proof set A = {v + u : u in W};
thus 0.V in v + W implies v in W
proof assume 0.V in v + W;
then 0.V in A by Def5;
then consider u such that A1: 0.V = v + u and A2: u in W;
v = - u by A1,RLVECT_1:def 10;
hence thesis by A2,Th30;
end;
assume v in W;
then A3: - v in W by Th30;
0.V = v - v by RLVECT_1:28
.= v + (- v) by RLVECT_1:def 11;
then 0.V in A by A3;
hence thesis by Def5;
end;

theorem Th59:
v in v + W
proof v + 0.V = v & 0.V in W by Th25,RLVECT_1:10;
then v in {v + u : u in W};
hence thesis by Def5;
end;

theorem
0.V + W = the carrier of W by Lm2;

theorem Th61:
v + (0).V = {v}
proof set A = {v + u : u in (0).V};
thus v + (0).V c= {v}
proof let x be set;
assume x in v + (0).V;
then x in A by Def5;
then consider u such that A1: x = v + u and A2: u in (0).V;
the carrier of (0).V = {0.V} & u in the carrier of (0).V
by A2,Def3,RLVECT_1:def 1
;
then u = 0.V by TARSKI:def 1;
then x = v by A1,RLVECT_1:10;
hence thesis by TARSKI:def 1;
end;
let x be set;
assume x in {v};
then A3: x = v by TARSKI:def 1;
0.V in (0).V & v = v + 0.V by Th25,RLVECT_1:10;
then x in A by A3;
hence thesis by Def5;
end;

Lm3: v in W iff v + W = the carrier of W
proof set A = {v + u : u in W};
thus v in W implies v + W = the carrier of W
proof assume A1: v in W;
thus v + W c= the carrier of W
proof let x be set;
assume x in v + W;
then x in A by Def5;
then consider u such that A2: x = v + u and A3: u in W;
v + u in W by A1,A3,Th28;
hence thesis by A2,RLVECT_1:def 1;
end;
let x be set;
assume x in the carrier of W;
then reconsider y = x, z = v as Element of W
by A1,RLVECT_1:def
1;

reconsider y1 = y, z1 = z as VECTOR of V by Th18;
A4: y - z in W by RLVECT_1:def 1;
A5: z + (y - z) = (y + z) - z by RLVECT_1:42
.= y + (z - z) by RLVECT_1:42
.= y + 0.W by RLVECT_1:28
.= x by RLVECT_1:10;
A6: y - z = y1 - z1 by Th24;
A7: y1 - z1 in W by A4,Th24;
z1 + (y1 - z1) = x by A5,A6,Th21;
then x in A by A7;
hence thesis by Def5;
end;
assume A8: v + W = the carrier of W;
assume A9: not v in W;
0.V in W & v + 0.V = v by Th25,RLVECT_1:10;
then v in {v + u : u in W};
then v in the carrier of W by A8,Def5;
hence thesis by A9,RLVECT_1:def 1;
end;

theorem Th62:
v + (Omega).V = the carrier of V
proof
A1: the carrier of (Omega).V = the carrier of the RLSStruct of V by Def4
.= the carrier of V;
then v in (Omega).V by RLVECT_1:def 1;
hence thesis by A1,Lm3;
end;

theorem Th63:
0.V in v + W iff v + W = the carrier of W
proof
(0.V in v + W iff v in W) & (v in
W iff v + W = the carrier of W) by Lm3,Th58;
hence thesis;
end;

theorem
v in W iff v + W = the carrier of W by Lm3;

theorem Th65:
v in W implies (a * v) + W = the carrier of W
proof set A = {a * v + u : u in W};
assume A1: v in W;
thus (a * v) + W c= the carrier of W
proof let x be set;
assume x in (a * v) + W;
then x in A by Def5;
then consider u such that A2: x = a * v + u and A3: u in W;
a * v in W by A1,Th29;
then a * v + u in W by A3,Th28;
hence thesis by A2,RLVECT_1:def 1;
end;
let x be set;
assume A4: x in the carrier of W;
the carrier of W c= the carrier of V & v in V by Def2,RLVECT_1:3;
then reconsider y = x as Element of V by A4;

a * v in W & x in W by A1,A4,Th29,RLVECT_1:def 1;
then A5: y - a * v in W by Th31;
a * v + (y - a * v) = (y + a * v) - a * v by RLVECT_1:42
.= y + (a * v - a * v) by RLVECT_1:42
.= y + 0.V by RLVECT_1:28
.= x by RLVECT_1:10;
then x in A by A5;
hence thesis by Def5;
end;

theorem Th66:
a <> 0 & (a * v) + W = the carrier of W implies v in W
proof assume that A1: a <> 0 and A2: (a * v) + W = the carrier of W;
assume not v in W;
then not 1 * v in W by RLVECT_1:def 9;
then not (a" * a) * v in W by A1,XCMPLX_0:def 7;
then not a" * (a * v) in W by RLVECT_1:def 9;
then A3: not a * v in W by Th29;
0.V in W & a * v + 0.V = a * v by Th25,RLVECT_1:10;
then a * v in {a * v + u : u in W};
then a * v in the carrier of W by A2,Def5;
end;

theorem Th67:
v in W iff - v + W = the carrier of W
proof
(v in W iff ((- 1) * v) + W = the carrier of W) & (- 1) * v = - v
by Th65,Th66,RLVECT_1:29;
hence thesis;
end;

theorem Th68:
u in W iff v + W = (v + u) + W
proof
set A = {v + v1 : v1 in W};
set B = {(v + u) + v2 : v2 in W};
thus u in W implies v + W = (v + u) + W
proof assume A1: u in W;
thus v + W c= (v + u) + W
proof let x be set;
assume x in v + W;
then x in A by Def5;
then consider v1 such that A2: x = v + v1 and A3: v1 in W;
A4: v1 - u in W by A1,A3,Th31;
(v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def 6
.= v + ((v1 + u) - u) by RLVECT_1:42
.= v + (v1 + (u - u)) by RLVECT_1:42
.= v + (v1 + 0.V) by RLVECT_1:28
.= x by A2,RLVECT_1:10;
then x in B by A4;
hence thesis by Def5;
end;
let x be set;
assume x in (v + u) + W;
then x in B by Def5;
then consider v2 such that A5: x = (v + u) + v2 and A6: v2 in W;
A7: u + v2 in W by A1,A6,Th28;
x = v + (u + v2) by A5,RLVECT_1:def 6;
then x in A by A7;
hence thesis by Def5;
end;
assume A8: v + W = (v + u) + W;
0.V in W & v + 0.V = v by Th25,RLVECT_1:10;
then v in A;
then v in (v + u) + W by A8,Def5;
then v in B by Def5;
then consider u1 such that A9: v = (v + u) + u1 and A10: u1 in W;
v = v + 0.V & v = v + (u + u1) by A9,RLVECT_1:10,def 6;
then A11:    u + u1 = 0.V by RLVECT_1:21;
u = - u1 by A11,RLVECT_1:def 10;
hence thesis by A10,Th30;
end;

theorem
u in W iff v + W = (v - u) + W
proof
A1: (- u in W iff v + W = (v + (- u)) + W) & v + (- u) = v - u
by Th68,RLVECT_1:def
11;
- u in W implies u in W
proof assume - u in W;
then - (- u) in W by Th30;
hence thesis by RLVECT_1:30;
end;
hence thesis by A1,Th30;
end;

theorem Th70:
v in u + W iff u + W = v + W
proof set A = {u + v1 : v1 in W}; set B = {v + v2 : v2 in W};
thus v in u + W implies u + W = v + W
proof assume v in u + W;
then v in A by Def5;
then consider z being VECTOR of V such that A1: v = u + z and
A2: z in W;
thus u + W c= v + W
proof let x be set;
assume x in u + W;
then x in A by Def5;
then consider v1 such that A3: x = u + v1 and A4: v1 in W;
A5: v1 - z in W by A2,A4,Th31;
v - z = u + (z - z) by A1,RLVECT_1:42
.= u + 0.V by RLVECT_1:28
.= u by RLVECT_1:10;
then x = (v + (- z)) + v1 by A3,RLVECT_1:def 11
.= v + (v1 + (- z)) by RLVECT_1:def 6
.= v + (v1 - z) by RLVECT_1:def 11;
then x in B by A5;
hence thesis by Def5;
end;
let x be set;
assume x in v + W;
then x in B by Def5;
then consider v2 such that A6: x = v + v2 and A7: v2 in W;
A8: z + v2 in W by A2,A7,Th28;
x = u + (z + v2) by A1,A6,RLVECT_1:def 6;
then x in A by A8;
hence thesis by Def5;
end;
thus thesis by Th59;
end;

theorem Th71:
v + W = (- v) + W iff v in W
proof
thus v + W = (- v) + W implies v in W
proof assume v + W = (- v) + W;
then v in (- v) + W by Th59;
then v in {- v + u : u in W} by Def5;
then consider u such that A1: v = - v + u and A2: u in W;
0.V = v - (- v + u) by A1,RLVECT_1:28
.= (v - (- v)) - u by RLVECT_1:41
.= (v + (- (- v))) - u by RLVECT_1:def 11
.= (v + v) - u by RLVECT_1:30
.= (1 * v + v) - u by RLVECT_1:def 9
.= (1 * v + 1 * v) - u by RLVECT_1:def 9
.= ((1 + 1) * v) - u by RLVECT_1:def 9
.= 2 * v - u;
then 2" * (2 * v) = 2" * u by RLVECT_1:35;
then (2" * 2) * v = 2" * u & 0 <> 2 by RLVECT_1:def 9;
then v = 2" * u by RLVECT_1:def 9;
hence thesis by A2,Th29;
end;
assume v in W;
then v + W = the carrier of W & (- v) + W = the carrier of W
by Lm3,Th67;
hence thesis;
end;

theorem Th72:
u in v1 + W & u in v2 + W implies v1 + W = v2 + W
proof assume that A1: u in v1 + W and A2: u in v2 + W;
set A = {v1 + u1 : u1 in W};
set B = {v2 + u2 : u2 in W};
u in A by A1,Def5;
then consider x1 being VECTOR of V such that A3: u = v1 + x1 and A4: x1 in
W;
u in B by A2,Def5;
then consider x2 being VECTOR of V such that A5: u = v2 + x2 and A6: x2 in
W;
thus v1 + W c= v2 + W
proof let x be set;
assume x in v1 + W;
then x in A by Def5;
then consider u1 such that A7: x = v1 + u1 and A8: u1 in W;
u - x1 = v1 + (x1 - x1) by A3,RLVECT_1:42
.= v1 + 0.V by RLVECT_1:28
.= v1 by RLVECT_1:10;
then A9: x = (v2 + (x2 - x1)) + u1 by A5,A7,RLVECT_1:42
.= v2 + ((x2 - x1) + u1) by RLVECT_1:def 6;
x2 - x1 in W by A4,A6,Th31;
then (x2 - x1) + u1 in W by A8,Th28;
then x in B by A9;
hence thesis by Def5;
end;
let x be set;
assume x in v2 + W;
then x in B by Def5;
then consider u1 such that A10: x = v2 + u1 and A11: u1 in W;
u - x2 = v2 + (x2 - x2) by A5,RLVECT_1:42
.= v2 + 0.V by RLVECT_1:28
.= v2 by RLVECT_1:10;
then A12: x = (v1 + (x1 - x2)) + u1 by A3,A10,RLVECT_1:42
.= v1 + ((x1 - x2) + u1) by RLVECT_1:def 6;
x1 - x2 in W by A4,A6,Th31;
then (x1 - x2) + u1 in W by A11,Th28;
then x in A by A12;
hence thesis by Def5;
end;

theorem
u in v + W & u in (- v) + W implies v in W
proof assume u in v + W & u in (- v) + W;
then v + W = (- v) + W by Th72;
hence thesis by Th71;
end;

theorem Th74:
a <> 1 & a * v in v + W implies v in W
proof
assume that A1: a <> 1 and A2: a * v in v + W;
A3: now assume a - 1 = 0;
then (- 1) + a = 0 by XCMPLX_0:def 8;
then a = - (- 1) by XCMPLX_0:def 6;
end;
a * v in {v + u : u in W} by A2,Def5;
then consider u such that A4: a * v = v + u and A5: u in W;
u = u + 0.V by RLVECT_1:10
.= u + (v - v) by RLVECT_1:28
.= a * v - v by A4,RLVECT_1:42
.= a * v - 1 * v by RLVECT_1:def 9
.= (a - 1) * v by RLVECT_1:49;
then (a - 1)" * u = ((a - 1)" * (a - 1)) * v & a - 1 <> 0
by A3,RLVECT_1:def 9;
then 1 * v = (a - 1)" * u by XCMPLX_0:def 7;
then v = (a - 1)" * u by RLVECT_1:def 9;
hence thesis by A5,Th29;
end;

theorem Th75:
v in W implies a * v in v + W
proof assume A1: v in W;
A2: a * v = (a - (1 - 1)) * v
.= ((a - 1) + 1) * v by XCMPLX_1:37
.= (a - 1) * v + 1 * v by RLVECT_1:def 9
.= v + (a - 1) * v by RLVECT_1:def 9;
(a - 1) * v in W by A1,Th29;
then a * v in {v + u : u in W} by A2;
hence thesis by Def5;
end;

theorem
- v in v + W iff v in W
proof
(v in W implies (- 1) * v in v + W) & (- 1) * v = - v &
(-1 <> 1 & (- 1) * v in v + W implies v in W) by Th74,Th75,RLVECT_1:29;
hence thesis;
end;

theorem Th77:
u + v in v + W iff u in W
proof set A = {v + v1 : v1 in W};
thus u + v in v + W implies u in W
proof assume u + v in v + W;
then u + v in A by Def5;
then consider v1 such that A1: u + v = v + v1 and A2: v1 in W;
thus thesis by A1,A2,RLVECT_1:21;
end;
assume u in W;
then u + v in A;
hence thesis by Def5;
end;

theorem
v - u in v + W iff u in W
proof
A1: v - u = (- u) + v by RLVECT_1:def 11;
A2: u in W implies - u in W by Th30;
- u in W implies - (- u) in W by Th30;
hence thesis by A1,A2,Th77,RLVECT_1:30;
end;

theorem Th79:
u in v + W iff
(ex v1 st v1 in W & u = v + v1)
proof set A = {v + v2 : v2 in W};
thus u in v + W implies (ex v1 st v1 in W & u = v + v1)
proof assume u in v + W;
then u in A by Def5;
then ex v1 st u = v + v1 & v1 in W;
hence thesis;
end;
given v1 such that A1: v1 in W & u = v + v1;
u in A by A1;
hence thesis by Def5;
end;

theorem
u in v + W iff
(ex v1 st v1 in W & u = v - v1)
proof set A = {v + v2 : v2 in W};
thus u in v + W implies (ex v1 st v1 in W & u = v - v1)
proof assume u in v + W;
then u in A by Def5;
then consider v1 such that A1: u = v + v1 and A2: v1 in W;
take x = - v1;
thus x in W by A2,Th30;
u = v + (- (- v1)) by A1,RLVECT_1:30
.= v - (- v1) by RLVECT_1:def 11;
hence thesis;
end;
given v1 such that A3: v1 in W & u = v - v1;
u = v + (- v1) & - v1 in W by A3,Th30,RLVECT_1:def 11;
then u in A;
hence thesis by Def5;
end;

theorem Th81:
(ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W
proof
thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W
proof given v such that A1: v1 in v + W and A2: v2 in v + W;
consider u1 such that A3: u1 in W and A4: v1 = v + u1 by A1,Th79;
consider u2 such that A5: u2 in W and A6: v2 = v + u2 by A2,Th79;
v1 - v2 = (u1 + v) + (- (v + u2)) by A4,A6,RLVECT_1:def 11
.= (u1 + v) + ((- v) - u2) by RLVECT_1:44
.= ((u1 + v) + (- v)) - u2 by RLVECT_1:42
.= (u1 + (v + (- v))) - u2 by RLVECT_1:def 6
.= (u1 + 0.V) - u2 by RLVECT_1:16
.= u1 - u2 by RLVECT_1:10;
hence thesis by A3,A5,Th31;
end;
assume v1 - v2 in W;
then A7: - (v1 - v2) in W by Th30;
take v1;
thus v1 in v1 + W by Th59;
v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by RLVECT_1:47
.= (v1 + (- v1)) + v2 by RLVECT_1:def 6
.= 0.V + v2 by RLVECT_1:16
.= v2 by RLVECT_1:10;
hence thesis by A7,Th79;
end;

theorem Th82:
v + W = u + W implies
(ex v1 st v1 in W & v + v1 = u)
proof
assume A1: v + W = u + W;
take v1 = u - v;
v in u + W by A1,Th59;
then v in {u + u2 : u2 in W} by Def5;
then consider u1 such that A2: v = u + u1 and A3: u1 in W;
0.V = (u + u1) - v by A2,RLVECT_1:28
.= u + (u1 - v) by RLVECT_1:42
.= u + ((- v) + u1) by RLVECT_1:def 11
.= (u + (- v)) + u1 by RLVECT_1:def 6
.= u1 + (u - v) by RLVECT_1:def 11;
then v1 = - u1 by RLVECT_1:def 10;
hence v1 in W by A3,Th30;
thus v + v1 = (u + v) - v by RLVECT_1:42
.= u + (v - v) by RLVECT_1:42
.= u + 0.V by RLVECT_1:28
.= u by RLVECT_1:10;
end;

theorem Th83:
v + W = u + W implies
(ex v1 st v1 in W & v - v1 = u)
proof
assume A1: v + W = u + W;
take v1 = v - u;
u in v + W by A1,Th59;
then u in {v + u2 : u2 in W} by Def5;
then consider u1 such that A2: u = v + u1 and A3: u1 in W;
0.V = (v + u1) - u by A2,RLVECT_1:28
.= v + (u1 - u) by RLVECT_1:42
.= v + ((- u) + u1) by RLVECT_1:def 11
.= (v + (- u)) + u1 by RLVECT_1:def 6
.= u1 + (v - u) by RLVECT_1:def 11;
then v1 = - u1 by RLVECT_1:def 10;
hence v1 in W by A3,Th30;
thus v - v1 = (v - v) + u by RLVECT_1:43
.= 0.V + u by RLVECT_1:28
.= u by RLVECT_1:10;
end;

theorem Th84:
for W1,W2 being strict Subspace of V holds
v + W1 = v + W2 iff W1 = W2
proof let W1,W2 be strict Subspace of V;
thus v + W1 = v + W2 implies W1 = W2
proof assume A1: v + W1 = v + W2;
the carrier of W1 = the carrier of W2
proof A2: the carrier of W1 c= the carrier of V by Def2;
A3: the carrier of W2 c= the carrier of V by Def2;
thus the carrier of W1 c= the carrier of W2
proof let x be set;
assume A4: x in the carrier of W1;
then reconsider y = x as Element of V by A2;

set z = v + y;
x in W1 by A4,RLVECT_1:def 1;
then z in {v + u : u in W1};
then z in v + W2 by A1,Def5;
then z in {v + u : u in W2} by Def5;
then consider u such that A5: z = v + u and A6: u in W2;
y = u by A5,RLVECT_1:21;
hence thesis by A6,RLVECT_1:def 1;
end;
let x be set;
assume A7: x in the carrier of W2;
then reconsider y = x as Element of V by A3;

set z = v + y;
x in W2 by A7,RLVECT_1:def 1;
then z in {v + u : u in W2};
then z in v + W1 by A1,Def5;
then z in {v + u : u in W1} by Def5;
then consider u such that A8: z = v + u and A9: u in W1;
y = u by A8,RLVECT_1:21;
hence thesis by A9,RLVECT_1:def 1;
end;
hence thesis by Th38;
end;
thus thesis;
end;

theorem Th85:
for W1,W2 being strict Subspace of V holds
v + W1 = u + W2 implies W1 = W2
proof let W1,W2 be strict Subspace of V;
assume A1: v + W1 = u + W2;
set V1 = the carrier of W1; set V2 = the carrier of W2;
assume A2: W1 <> W2;
then V1 <> V2 by Th38;
then A3: not V1 c= V2 or not V2 c= V1 by XBOOLE_0:def 10;

A4: now assume A5: V1 \ V2 <> {};
consider x being Element of V1 \ V2;
x in V1 & not x in V2 by A5,XBOOLE_0:def 4;
then A6: x in W1 & not x in W2 by RLVECT_1:def 1;
then x in V by Th17;
then reconsider x as Element of V by RLVECT_1:def 1;

set z = v + x;
z in {v + u2 : u2 in W1} by A6;
then z in u + W2 by A1,Def5;
then z in {u + u2 : u2 in W2} by Def5;
then consider u1 such that A7: z = u + u1 and A8: u1 in W2;
x = 0.V + x by RLVECT_1:10
.= v - v + x by RLVECT_1:28
.= (- v + v) + x by RLVECT_1:def 11
.= - v + (u + u1) by A7,RLVECT_1:def 6;
then A9: (v + (- v + (u + u1))) + W1 = v + W1 by A6,Th68;
v + (- v + (u + u1)) = (v + (- v)) + (u + u1) by RLVECT_1:def 6
.= (v - v) + (u + u1) by RLVECT_1:def 11
.= 0.V + (u + u1) by RLVECT_1:28
.= u + u1 by RLVECT_1:10;
then (u + u1) + W2 = (u + u1) + W1 by A1,A8,A9,Th68;
hence thesis by A2,Th84;
end;
now assume A10: V2 \ V1 <> {};
consider x being Element of V2 \ V1;
x in V2 & not x in V1 by A10,XBOOLE_0:def 4;
then A11: x in W2 & not x in W1 by RLVECT_1:def 1;
then x in V by Th17;
then reconsider x as Element of V by RLVECT_1:def 1;

set z = u + x;
z in {u + u2 : u2 in W2} by A11;
then z in v + W1 by A1,Def5;
then z in {v + u2 : u2 in W1} by Def5;
then consider u1 such that A12: z = v + u1 and A13: u1 in W1;
x = 0.V + x by RLVECT_1:10
.= u - u + x by RLVECT_1:28
.= (- u + u) + x by RLVECT_1:def 11
.= - u + (v + u1) by A12,RLVECT_1:def 6;
then A14: (u + (- u + (v + u1))) + W2 = u + W2 by A11,Th68;
u + (- u + (v + u1)) = (u + (- u)) + (v + u1) by RLVECT_1:def 6
.= (u - u) + (v + u1) by RLVECT_1:def 11
.= 0.V + (v + u1) by RLVECT_1:28
.= v + u1 by RLVECT_1:10;
then (v + u1) + W1 = (v + u1) + W2 by A1,A13,A14,Th68;
hence thesis by A2,Th84;
end;
hence thesis by A3,A4,XBOOLE_1:37;
end;

::
::  Theorems concerning cosets of subspace
::  regarded as subsets of the carrier.
::

theorem
C is lineary-closed iff C = the carrier of W
proof
thus C is lineary-closed implies C = the carrier of W
proof assume A1: C is lineary-closed;
consider v such that A2: C = v + W by Def6;
C <> {} by A2,Th59;
then 0.V in v + W by A1,A2,Th4;
hence thesis by A2,Th63;
end;
thus thesis by Lm1;
end;

theorem
for W1,W2 being strict Subspace of V,
C1 being Coset of W1, C2 being Coset of W2 holds
C1 = C2 implies W1 = W2
proof
let W1,W2 be strict Subspace of V,
C1 be Coset of W1, C2 be Coset of W2;
A1:    ex v1 st C1 = v1 + W1 by Def6;
ex v2 st C2 = v2 + W2 by Def6;
hence thesis by A1,Th85;
end;

theorem
{v} is Coset of (0).V
proof v + (0).V = {v} by Th61;
hence thesis by Def6;
end;

theorem
V1 is Coset of (0).V implies (ex v st V1 = {v})
proof assume V1 is Coset of (0).V;
then consider v such that A1: V1 = v + (0).V by Def6;
take v;
thus thesis by A1,Th61;
end;

theorem
the carrier of W is Coset of W
proof the carrier of W = 0.V + W by Lm2;
hence thesis by Def6;
end;

theorem
the carrier of V is Coset of (Omega).V
proof
the carrier of V is Subset of V iff
the carrier of V c= the carrier of V;
then reconsider A = the carrier of V as Subset of V;
consider v;
A = v + (Omega).V by Th62;
hence thesis by Def6;
end;

theorem
V1 is Coset of (Omega).V implies V1 = the carrier of V
proof assume V1 is Coset of (Omega).V;
then ex v st V1 = v + (Omega).V by Def6;
hence thesis by Th62;
end;

theorem
0.V in C iff C = the carrier of W
proof
ex v st C = v + W by Def6;
hence thesis by Th63;
end;

theorem Th94:
u in C iff C = u + W
proof
thus u in C implies C = u + W
proof assume A1: u in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th70;
end;
thus thesis by Th59;
end;

theorem
u in C & v in C implies (ex v1 st v1 in W & u + v1 = v)
proof assume u in C & v in C;
then C = u + W & C = v + W by Th94;
hence thesis by Th82;
end;

theorem
u in C & v in C implies (ex v1 st v1 in W & u - v1 = v)
proof assume u in C & v in C;
then C = u + W & C = v + W by Th94;
hence thesis by Th83;
end;

theorem
(ex C st v1 in C & v2 in C) iff v1 - v2 in W
proof
thus (ex C st v1 in C & v2 in C) implies v1 - v2 in W
proof given C such that A1: v1 in C & v2 in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th81;
end;
assume v1 - v2 in W;
then consider v such that A2: v1 in v + W & v2 in v + W by Th81;
reconsider C = v + W as Coset of W by Def6;
take C;
thus thesis by A2;
end;

theorem
u in B & u in C implies B = C
proof assume A1: u in B & u in C;
A2:    ex v1 st B = v1 + W by Def6;
ex v2 st C = v2 + W by Def6;
hence thesis by A1,A2,Th72;
end;
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